GRAVITATION

CONTENTS

1. Geometrodynamics in Brief ..... 3
2. The Parable of the Apple ..... 3
3. Spacetime With and Without Coordinates ..... 5
4. Weightlessness ..... 13
5. Local Lorentz Geometry, With and Without Coordinates ..... 19
   Time ..... 23
6. Curvature ..... 29
   Effect of Matter on Geometry ..... 37
   Part II PHYSICS IN FLAT SPACETIME ..... 45
7. Foundations of Special Relativity ..... 47
8. Overview ..... 47
9. Geometric Objects ..... 48
10. Vectors ..... 49
11. The Metric Tensor ..... 51
12. Differential Forms ..... 53
13. Gradients and Directional Derivatives ..... 59
14. Coordinate Representation of Geometric Objects ..... 60 ..... 63
15. The Centrifuge and the Photon
16. Lorentz Transformations ..... 66
    $\stackrel{\rightharpoonup }{0}$$\stackrel{\rightharpoonup }{0}$0^(⇀)\stackrel{\rightharpoonup}{0}$\stackrel{\rightharpoonup }{0}$ Collisions ..... 69
17. The Electromagnetic Field ..... 71
18. The Lorentz Force and the Electromagnetic Field Tensor ..... 71
19. Tensors in All Generality ..... 74
20. Three-Plus-One View Versus Geometric View ..... 78
21. Maxwell's Equations ..... 79
    Working with Tensors ..... 81
22. Electromagnetism and Differential Forms ..... 90
23. Exterior Calculus ..... 90
24. Electromagnetic 2-Form and Lorentz Force ..... 99
25. Forms Illuminate Electromagnetism and Electromagnetism Illuminates Forms ..... 105
26. Radiation Fields ..... 110
27. Maxwell's Equations ..... 112
28. Exterior Derivative and Closed Forms ..... 114
29. Distant Action from Local Law ..... 120
30. Stress-Energy Tensor and Conservation Laws ..... 130
31. Track-1 Overview ..... 130
32. Three-Dimensional Volumes and Definition of the Stress-Energy Tensor ..... 130
33. Components of Stress-Energy Tensor ..... 137
34. Stress-Energy Tensor for a Swarm of Particles ..... 138
35. Stress-Energy Tensor for a Perfect Fluid ..... 139
36. Electromagnetic Stress-Energy ..... 140
37. Symmetry of the Stress-Energy Tensor ..... 141
38. Conservation of 4-Momentum: Integral Formulation ..... 142
39. Conservation of 4-Momentum: Differential Formulation ..... 146
40. Sample Application of $\mathbf{\nabla }\cdot \mathit{T}=0$$\mathbf{\nabla }\cdot \mathit{T}=0$grad*T=0\boldsymbol{\nabla} \cdot \boldsymbol{T}=0$\mathbf{\nabla }\cdot \mathit{T}=0$ ..... 152
41. Angular Momentum ..... 156
42. Accelerated Observers ..... 163
43. Accelerated Observers Can Be Analyzed Using Special Relativity ..... 163
44. Hyperbolic Motion ..... 166
45. Constraints on Size of an Accelerated Frame ..... 168
46. The Tetrad Carried by a Uniformly Accelerated Observer ..... 169
47. The Tetrad Fermi-Walker Transported by an Observer with ArbitraryAcceleration 170
48. The Local Coordinate System of an Accelerated Observer ..... 172
49. Incompatibility of Gravity and Special Relativity ..... 177
50. Attempts to Incorporate Gravity into Special Relativity ..... 177
51. Gravitational Redshift Derived from Energy Conservation ..... 187
52. Gravitational Redshift Implies Spacetime Is Curved ..... 187
53. Gravitational Redshift as Evidence for the Principle of Equivalence ..... 189
54. Local Flatness, Global Curvature ..... 190
    Part III THE MATHEMATICS OF CURVED SPACETIME ..... 193
55. Differential Geometry: An Overview ..... 195
    An Overview of Part III 195
56. 2. Track 1 Versus Track 2: Difference in Outlook and Power ..... 197

1. Geometric Objects in Metric-Free, Geodesic-Free Spacetime ..... 225
2. "Vector" and "Directional Derivative" Refined into Tangent Vector ..... 226
3. Bases, Components, and Transformation Laws for Vectors ..... 230
4. 1-Forms ..... 231
5. Tensors ..... 233
6. Commutators and Pictorial Techniques ..... 235
7. Manifolds and Differential Topology ..... 240
8. Affine Geometry: Geodesics, Parallel Transport and Covariant Derivative ..... 244
9. Geodesics and the Equivalence Principle ..... 244
10. Parallel Transport and Covariant Derivative: Pictorial Approach ..... 245
11. Parallel Transport and Covariant Derivative: Abstract Approach ..... 247
12. Parallel Transport and Covariant Derivative: Component Approach ..... 258
13. Geodesic Equation ..... 262
14. Geodesic Deviation and Spacetime Curvature ..... 265
15. Curvature, At Last! ..... 265
16. The Relative Acceleration of Neighboring Geodesics ..... 265
17. Tidal Gravitational Forces and Riemann Curvature Tensor ..... 270
18. Parallel Transport Around a Closed Curve ..... 277
19. Flatness is Equivalent to Zero Riemann Curvature ..... 283
20. Riemann Normal Coordinates ..... 285
21. Newtonian Gravity in the Language of Curved Spacetime ..... 289
22. Newtonian Gravity in Brief ..... 289
23. Stratification of Newtonian Spacetime ..... 291
24. Galilean Coordinate Systems ..... 292
25. Geometric, Coordinate-Free Formulation of Newtonian Gravity ..... 298
26. The Geometric View of Physics: A Critique ..... 302
27. Riemannian Geometry: Metric as Foundation of All ..... 304
28. New Features Imposed on Geometry by Local Validity of Special Relativity 304
29. Metric ..... 305
30. Concord Between Geodesics of Curved Spacetime Geometry and Straight Lines of Local Lorentz Geometry ..... 312
31. Geodesics as World Lines of Extremal Proper Time ..... 315
32. Metric-Induced Properties of Riemann ..... 324
33. The Proper Reference Frame of an Accelerated Observer ..... 327
34. Calculation of Curvature ..... 333
35. Curvature as a Tool for Understanding Physics ..... 333
36. Forming the Einstein Tensor ..... 343
37. More Efficient Computation ..... 344
38. The Geodesic Lagrangian Method ..... 344
39. Curvature 2-Forms ..... 348
40. Computation of Curvature Using Exterior Differential Forms ..... 354
41. Bianchi Identities and the Boundary of a Boundary ..... 364
42. Bianchi Identities in Brief ..... 364
43. Bianchi Identity $\mathit{d}\mathrm{\%}=0$$\mathit{d}\mathrm{\%}=0$d%=0\boldsymbol{d} \%=0$\mathit{d}\mathrm{\%}=0$ as a Manifestation of "Boundary ofBoundary $=0$$=0$=0=0$=0$ "372
44. Moment of Rotation: Key to Contracted Bianchi Identity ..... 373
45. Calculation of the Moment of Rotation ..... 375
46. Conservation of Moment of Rotation Seen from "Boundary of a Boundary is Zero" 377
47. Conservation of Moment of Rotation Expressed in Differential Form ..... 378
48. From Conservation of Moment of Rotation to Einstein's Geometrodynamics: A Preview ..... 379
    Part IV EINSTEIN'S GEOMETRIC THEORY OF GRAVITY ..... 383
49. Equivalence Principle and Measurement of the "Gravitational Field" ..... 385
50. Overview ..... 385
    The Laws of Physics in Curved Spacetime ..... 385
    Factor-Ordering Problems in the Equivalence Principle ..... 388
51. The Rods and Clocks Used to Measure Space and Time Intervals ..... 393
    The Measurement of the Gravitational Field ..... 399
52. How Mass-Energy Generates Curvature ..... 404
53. Automatic Conservation of thof the Field Equation 404
54. Automatic Conservation of the Source: A Dynamic Necessity ..... 408 ..... 409
    Cosmological Constant
    Cosmological Constant
55. The Newtonian Limit ..... 412
    Axiomatize Einstein's Theory? ..... 416
56. "No Prior Geometry": A Feature Distinguishing Einstein's Theory from Other Theories of Gravity ..... 429
57. A Taste of the History of Einstein's Equation ..... 431
58. Weak Gravitational Fields ..... 435
59. The Linearized Theory of Gravity ..... 435
    Gravitational Waves ..... 442
60. Effect of Gravity on Matter ..... 442
    Nearly Newtonian Gravitational Fields ..... 445
61. Mass and Angular Momentum of a Gravitating System ..... 448
62. External Field of a Weakly Gravitating Source ..... 448
63. Measurement of the Mass and Angular Momentum ..... 450
64. Mass and Angular Momentum of Fully Relativistic Sources ..... 451
65. Mass and Angular Momentum of a Closed Universe ..... 457
66. Conservation Laws for 4-Momentum and Angular Momentum ..... 460
67. Overview ..... 460
68. Gaussian Flux Integrals for 4-Momentum and Angular Momentum ..... 461
69. Volume Integrals for 4-Momentum and Angular Momentum ..... 464
70. Why the Energy of the Gravitational Field Cannot be Localized ..... 466
71. Conservation Laws for Total 4-Momentum and Angular Momentum ..... 468
72. Equation of Motion Derived from the Field Equation ..... 471
73. Variational Principle and Initial-Value Data ..... 484
74. Dynamics Requires Initial-Value Data ..... 484
75. The Hilbert Action Principle and the Palatini Method of Variation ..... 491
76. Matter Lagrangian and Stress-Energy Tensor ..... 504
77. Splitting Spacetime into Space and Time ..... 505
78. Intrinsic and Extrinsic Curvature ..... 509
79. The Hilbert Action Principle and the Arnowitt-Deser-Misner Modification Thereof in the Space-plus-Time Split ..... 519
80. The Arnowitt-Deser-Misner Formulation of the Dynamics of Geometry ..... 520
81. Integrating Forward in Time ..... 526
82. The Initial-Value Problem in the Thin-Sandwich Formulation ..... 528
83. The Time-Symmetric and Time-Antisymmetric Initial-Value Problem ..... 535
84. York's "Handles" to Specify a 4-Geometry ..... 539
85. Mach's Principle and the Origin of Inertia ..... 543
86. Junction Conditions ..... 551

22. Thermodynamics, Hydrodynamics, Electrodynamics, Geometric Optics, and Kinetic Theory 557

1. The Why of this Chapter 557
2. Thermodynamics in Curved Spacetime 557
3. Hydrodynamics in Curved Spacetime 562
4. Electrodynamics in Curved Spacetime 568
5. Geometric Optics in Curved Spacetime 570
6. Kinetic Theory in Curved Spacetime 583

Part V RELATIVISTIC STARS 591

23. Spherical Stars 593
24. Prolog 593
25. Coordinates and Metric for a Static, Spherical System 594
26. Physical Interpretation of Schwarzschild coordinates 595
27. Description of the Matter Inside a Star 597
28. Equations of Structure 600
29. External Gravitational Field 607
30. How to Construct a Stellar Model 608
31. The Spacetime Geometry for a Static Star 612
32. Pulsars and Neutron Stars; Quasars and Supermassive Stars 618
33. Overview 618
34. The Endpoint of Stellar Evolution 621
35. Pulsars 627
36. Supermassive Stars and Stellar Instabilities 630
37. Quasars and Explosions In Galactic Nuclei 634
38. Relativistic Star Clusters 634
39. The "Pit in the Potential" as the Central New Feature of Motion in
    Schwarzschild Geometry 636
40. From Kepler's Laws to the Effective Potential for Motion in Schwarzschild Geometry 636
41. Symmetries and Conservation Laws 650
42. Conserved Quantities for Motion in Schwarzschild Geometry 655
43. Gravitational Redshift 659
44. Orbits of Particles 659
45. Orbit of a Photon, Neutrino, or Graviton in Schwarzschild Geometry 672
46. Spherical Star Clusters 679
47. Stellar Pulsations 688
48. Motivation 688
49. Setting Up the Problem 689
50. Eulerian versus Lagrangian Perturbations 690
51. Initial-Value Equations 691
52. Dynamic Equation and Boundary Conditions 693
53. Summary of Results 694

Part VI THE UNIVERSE
701

27. Idealized Cosmologies
703

1. The "Standard Model" of the Universe ..... 763
2. Standard Model Modified for Primordial Chaos ..... 769
3. What "Preceded" the Initial Singularity? ..... 769
4. Other Cosmological Theories ..... 770
5. Present State and Future Evolution of the Universe ..... 771
6. Parameters that Determine the Fate of the Universe ..... 771
7. Cosmological Redshift ..... 772
8. The Distance-Redshift Relation: Measurement of the Hubble Constant ..... 780
9. The Magnitude-Redshift Relation: Measurement of the Deceleration Parameter ..... 782
10. Search for "Lens Effect" of the Universe ..... 795
11. Density of the Universe Today ..... 796
12. Summary of Present Knowledge About Cosmological Parameters ..... 797
13. Anisotropic and Inhomogeneous Cosmologies ..... 800
14. Why Is the Universe So Homogeneous and Isotropic? ..... 800
15. The Kasner Model for an Anisotropic Universe ..... 801
16. Adiabatic Cooling of Anisotropy ..... 802
17. Viscous Dissipation of Anisotropy ..... 802
18. Particle Creation in an Anisotropic Universe ..... 803
19. Inhomogeneous Cosmologies ..... 804
20. The Mixmaster Universe ..... 805
21. Horizons and the Isotropy of the Microwave Background ..... 815
    Part VII GRAVITATIONAL COLLAPSE AND BLACK HOLES ..... 817
22. Schwarzschild Geometry ..... 819
23. Inevitability of Collapse for Massive Stars ..... 819
24. The Nonsingularity of the Gravitational Radius ..... 820
25. Behavior of Schwarzschild Coordinates at $r=2M$$r=2M$r=2Mr=2 M$r=2M$ ..... 823
26. Several Well-Behaved Coordinate Systems ..... 826
27. Relationship Between Kruskal-Szekeres Coordinates and Schwarzschild Coordinates ..... 833
28. Dynamics of the Schwarzschild Geometry ..... 836
29. Gravitational Collapse ..... 842
30. Relevance of Schwarzschild Geometry ..... 842
31. Birkhoff's Theorem ..... 843
32. Exterior Geometry of a Collapsing Star ..... 846
33. Collapse of a Star with Uniform Density and Zero Pressure ..... 851
34. Spherically Symmetric Collapse with Internal Pressure Forces ..... 857
35. The Fate of a Man Who Falls into the Singularity at $r=0$$r=0$r=0r=0$r=0$ ..... 860
36. Realistic Gravitational Collapse-An Overview ..... 862

33. Black Holes 872

1. Why "Black Hole"? 872
2. The Gravitational and Electromagnetic Fields of a Black Hole 875
3. Mass, Angular Momentum, Charge, and Magnetic Moment 891
4. Symmetries and Frame Dragging 892
5. Equations of Motion for Test Particles 897
6. Principal Null Congruences 901
7. Storage and Removal of Energy from Black Holes 904
8. Reversible and Irreversible Transformations 907
9. Global Techniques, Horizons, and Singularity Theorems 916
10. Global Techniques Versus Local Techniques 916
11. "Infinity" in Asymptotically Flat Spacetime 917
12. Causality and Horizons 922
13. Global Structure of Horizons 924
14. Proof of Second Law of Black-Hole Dynamics 931
15. Singularity Theorems and the "Issue of the Final State" 934

Part VIII GRAVITATIONAL WAVES 941

35. Propagation of Gravitational Waves 943

1. Viewpoints 943
2. Review of "Linearized Theory" in Vacuum 944
3. Plane-Wave Solutions in Linearized Theory 945
4. The Transverse Traceless (TT) Gauge 946
5. Geodesic Deviation in a Linearized Gravitational Wave 950

1. The Quadrupole Nature of Gravitational Waves 974
2. Power Radiated in Terms of Internal Power Flow 978
3. Laboratory Generators of Gravitational Waves 979
4. Astrophysical Sources of Gravitational Waves: General Discussion 980

1. Coordinate Systems and Impinging Waves 1004
2. Accelerations in Mechanical Detectors 1006
3. Types of Mechanical Detectors 1012
4. Vibrating, Mechanical Detectors: Introductory Remarks ..... 1019
5. Idealized Wave-Dominated Detector, Excited by Steady Flux of MonochromaticWaves1022
6. Idealized, Wave-Dominated Detector, Excited by Arbitrary Flux of Radiation ..... 1026
7. General Wave-Dominated Detector, Excited by Arbitrary Flux ofRadiation 1028
8. Noisy Detectors ..... 1036
9. Nonmechanical Detectors ..... 1040
10. Looking Toward the Future ..... 1040
    Part IX. EXPERIMENTAL TESTS OF GENERAL RELATIVITY ..... 1045
11. Testing the Foundations of Relativity ..... 1047
12. Testing is Easier in the Solar System than in Remote Space ..... 1047
    Theoretical Frameworks for Analyzing Tests of General Relativity ..... 1048
13. Tests of the Principle of the Uniqueness of Free Fall: Eötvös-Dicke Experiment ..... 1050
14. Tests for the Existence of a Metric Governing Length and Time Measurements ..... 1054
15. Tests of Geodesic Motion: Gravitational Redshift Experiments ..... 1055
16. Tests of the Equivalence Principle ..... 1060
17. Tests for the Existence of Unknown Long-Range Fields ..... 1063
18. Other Theories of Gravity and the Post-Newtonian Approximation ..... 1066
19. Other Theories ..... 1066
20. Metric Theories of Gravity ..... 1067
21. Post-Newtonian Limit and PPN Formalism ..... 1068
22. PPN Coordinate System ..... 1073
23. Description of the Matter in the Solar System ..... 1074
24. Nature of the Post-Newtonian Expansion ..... 1075
25. Newtonian Approximation ..... 1077
26. PPN Metric Coefficients ..... 1080
27. Velocity of PPN Coordinates Relative to "Universal Rest Frame" ..... 1083
28. PPN Stress-Energy Tensor ..... 1086
29. PPN Equations of Motion ..... 1087
30. Relation of PPN Coordinates to Surrounding Universe ..... 1091
31. Summary of PPN Formalism ..... 109
32. Solar-System Experiments ..... 1096
33. Many Experiments Open to Distinguish General Relativity from Proposed Metric Theories of Gravity 1096
34. The Use of Light Rays and Radio Waves to Test Gravity ..... 1099
35. "Light" Deflection ..... 1101
36. Time-Delay in Radar Propagation ..... 1103
37. Perihelion Shift and Periodic Perturbations in Geodesic Orbits ..... 1110
38. Three-Body Effects in the Lunar Orbit ..... 1116
39. The Dragging of Inertial Frames ..... 1117
40. Is the Gravitational Constant Constant? ..... 1121
41. Do Planets and the Sun Move on Geodesics? ..... 1126
42. Summary of Experimental Tests of General Relativity ..... 1131
    Part X. FRONTIERS ..... 1133
43. Spinors ..... 1135
44. Reflections, Rotations, and the Combination of Rotations ..... 1135
45. Infinitesimal Rotations ..... 1140
46. Lorentz Transformation via Spinor Algebra ..... 1142
47. Thomas Precession via Spinor Algebra ..... 1145
48. Spinors ..... 1148
49. Correspondence Between Vectors and Spinors ..... 1150
50. Spinor Algebra ..... 1151
51. Spin Space and Its Basis Spinors ..... 1156
52. Spinor Viewed as Flagpole Plus Flag Plus Orientation-Entanglement Relation ..... 1157
53. Appearance of the Night Sky: An Application of Spinors ..... 1160
54. Spinors as a Powerful Tool in Gravitation Theory ..... 1164
55. Regge Calculus ..... 1166
56. Why the Regge Calculus? ..... 1166
57. Regge Calculus in Brief ..... 1166
58. Simplexes and Deficit Angles ..... 1167
59. Skeleton Form of Field Equations ..... 1169
60. The Choice of Lattice Structure ..... 1173
61. The Choice of Edge Lengths ..... 1177
62. Past Applications of Regge Calculus ..... 1178
63. The Future of Regge Calculus ..... 1179
64. Superspace: Arena for the Dynamics of Geometry ..... 1180
65. Space, Superspace, and Spacetime Distinguished ..... 1180
66. The Dynamics of Geometry Described in the Language of the Superspace of the (3) ${}^{\prime }$${}^{\prime }$^('){ }^{\prime}${}^{\prime }$ 's ..... 1184
67. The Einstein-Hamilton-Jacobi Equation ..... 1185
68. Fluctuations in Geometry ..... 1190
69. Beyond the End of Time ..... 1196
70. Gravitational Collapse as the Greatest Crisis in Physics of All Time ..... 1196
71. Assessment of the Theory that Predicts Collapse ..... 1198
72. Vacuum Fluctuations: Their Prevalence and Final Dominance ..... 1202
73. Not Geometry, but Pregeometry, as the Magic Building Materia ..... 1203
74. Pregeometry as the Calculus of Propositions ..... 1208
75. The Black Box: The Reprocessing of the Universe ..... 1209
    Bibliography and Index of Names ..... 1221
    Subject Index ..... 1255

BOXES

4.1. Differential forms and exterior calculus in brief. 91

12.4. Geometric versus standard formulation of Newtonian gravity. 300

FIGURES

40.3. Coordinates for calculating the relativistic timedelay. 1106

41.1. Combination of rotations of ${90}^{\circ }$${90}^{\circ }$90^(@)90^{\circ}${90}^{\circ }$ about axes that

FOREWORD TO THE 2017 PRINTING OF GRAVITATION
DAVID I. KAISER

ACKNOWLEDGMENTS

PREFACE TO THE 2017 PRINTING OF GRAVITATION CHARLES W. MISNER AND KIP S. THORNE

the context in which we wrote gravitation

GRAVITATION'S GEOMETRIC VIEWPOINT

HOW USEFUL CAN GRAVITATION BE TODAY?

CHAPTER-BY-CHAPTER STATUS OF GRAVITATION

1. Parts I, II, III, and IV, the fundamentals of general relativity, have not changed significantly over the past 44 years, so Chapters 1-22 on the fundamentals are almost fully up to date. The only exceptions are the following.
   A. Chapter 8, Differential Geometry, should be augmented by an introduction to symbolic manipulation software (e.g., Maple, Mathematica, and Matlab) for computing connection coefficients and curvature tensors and performing other tensorial calculations; and Chapter 14, Calculation of Curvature, could be augmented by a deeper treatment of symbolic manipulation.
   B. To Part IV, Einstein's Geometric Theory of Gravity, we would add four new topics:
   a. Numerical relativity, which underpins gravitational wave observations and is teaching us about the nonlinear dynamics of curved spacetime; for example, Maggiore (2017), or for far greater detail, Baumgarte and Shapiro (2010) and Shibata (2016).
   b. Gravitational lensing, which is based on the linearized approximation to general relativity (Section 18.1) and has become a major tool for astronomy; for example, MCP or Straumann (2013), or for far greater detail, Schneider, Ehlers, and Falco (1992).
   c. The Einstein field equation in higher dimensions, particularly four space dimensions and one time dimension, which is motivated by string theory's requirement for higher dimensions and by the Randall-Sundrum (1999a,b) insight that one or more of these higher dimensions could be macroscopic. This topic often goes under the name "Braneworlds." For a brief treatment see, for example, Zee (2013); for much greater detail at the level of Gravitation, see Maartens and Koyama (2010).
   d. Quantum field theory in curved spacetime (which could be added at the end of Chapter 22). This topic underpins, most importantly, Hawking radiation from black holes; see below. For a brief introduction see, for example, Carroll (2004); for more thorough treatments, see Wald (1994) and Parker and Toms (2009).
2. Part V, Relativistic Stars, is similarly almost fully up to date, with the following two exceptions.
   A. Chapter 24, Pulsars and Neutron Stars; Quasars and Supermassive Stars, is completely out of date. Observations and observation-driven astrophysical theory have transformed our understanding profoundly. See, for example, Straumann (2013) or Maggiore (2017) or, for far greater detail, Shapiro and Teukolsky (1983), which is somewhat out of date but excellent and thorough.
   B. Chapter 25 on geodesic orbits in the Schwarzschild spacetime should be augmented by exercises on computing orbits numerically to give the reader physical insight-which is best done by numerically integrating the Hamilton equations that follow from the super-Hamiltonian (Exercise 25.2); see Levin and Perez-Giz (2008).
3. Part VI, The Universe, is for the most part tremendously out of date.
   A. Chapter 27, Idealized Cosmologies, is an exception. The fundamental ideas and equations for idealized cosmologies have not changed, but the emphasis of this chapter is archaic. John, our mentor-whose intuition and prescience
   were usually superb (Misner, Thorne, and Zurek 2009)-was firmly convinced that our universe would turn out to be closed and have vanishing cosmological constant; so in Gravitation, the closed Friedman cosmology is given great emphasis. Since Gravitation was published, a rich set of cosmological observations has revealed that our universe is very nearly flat spatially and has a positive cosmological constant (or something resembling it). So this chapter should be augmented by a more detailed treatment of the material in Section 27.11, and most importantly, by an in-depth treatment of the de Sitter solution of the Einstein equation with cosmological constant - as, for example, in Hawking and Ellis (1973). As a side issue (a Box), we would add the anti-de Sitter (AdS) solution (e.g., Hawking and Ellis 1973), because of its importance today in explorations of fundamental physics (e.g., the AdS/CFT correspondence).
   B. Chapter 28, Evolution of the Universe into Its Present State, and Chapter 29, Present State and Future Evolution of the Universe, are completely out of date and thus only of historic interest. During the past two decades, these subjects have been thoroughly transformed by cosmological observations and associated theory. For a fully up-to-date, pedagogical treatment, we recommend chapter 28 of MCP, or at a more elementary level, Schneider (2015). The most useful advanced textbook may be Weinberg (2008).
   C. Cosmological observations over the past two decades suggest that Chapter 30, Anisotropic and Homogeneous Cosmologies is likely not relevant to the early evolution of our universe. However, it is of great importance for a fundamental new topic to be discussed below (see 4.D): the physical structure of singularities.
   D. To this cosmological Part of Gravitation, we would add a major new topic (chapter): Inflationary expansion in the very early universe, as treated, for example, in Peacock (1999); Hobson, Efstathiou, and Lasenby (2006); Sasaki (2015); and section 28.7.1 of MCP.
4. Part VII, Gravitational Collapse and Black Holes, is surprisingly up to date, in large measure because it focuses on theory and says little about observations. However, a few new theoretical developments (some major) have emerged since 1973 that should be included in any year-long advanced course on general relativity.
   A. To Chapter 31, Schwarzschild Geometry, and Chapter 32, Gravitational Collapse, we would add nothing.
   B. To Chapter 33, Black Holes, we would add the following topics.
   a. Exercises to explore geodesic orbits around a Kerr black hole numerically, by integrating Hamilton's equations for the super-Hamiltonian (33.27c) (Levin and Perez-Giz 2008).
   b. A discussion of quasinormal modes of a Kerr black hole, motivated by Exercise 33.14; see, for example, chapter 12 of Maggiore (2017). (The first hint of these modes was found by Vishveshwara, 1970, in the form of ringdown waves like those that LIGO has detected 45 years later. By 1973, when Gravitation was published, the concept of quasinormal modes was fully in hand along with the equations for computing them, but the first numerical computation of their complex eigenfrequencies and eigenfunctions, by Chandrasekhar and Detweiler, 1975, was still two years in the future.)
   c. Spherical accretion onto a Schwarzschild black hole and accretion disks around a Kerr black hole: at least a few exercises as, for example, in MCP. These topics are touched on in Box 33.3 of Gravitation, but given their great astrophysical importance today, they deserve greater and more up-to-date detail; see, for example, the brief discussion in Straumann (2013), the longer discussion in Abramovicz and Fragile (2013), or the very detailed discussion in Meier (2012).
   d. The Blandford-Znajek (1977) mechanism by which magnetic fields extract spin energy from black holes to power jets; see, for example, MCP. For far greater detail, see Thorne, Price, and MacDonald (1986), which emphasizes the relativity, and McKinney, Tchekhovskoy, and Blandford (2012), which emphasizes the astrophysics.
   e. Some discussion of astronomical observations of black holes and their astrophysical roles in the universe; see, for example, Narayan and McClintock (2015) or for greater detail, Meier (2012) and Schneider (2015). (What remarkable developments there have been here, since Gravitation was published!)
   f. Hawking radiation, the associated thermal atmosphere of a black hole, and black-hole thermodynamics (all of which were developed within a year of publication of Gravitation, in the wake of Stephen Hawking's and others' bringing quantum field theory in curved spacetime into a sufficiently mature form; see 1.B.c above). See, for example, Carroll (2004) for a moderately brief, pedagogical treatment, and Wald (1994) for greater detail.
   C. To Chapter 34, Global Techniques, Horizons, and Singularity Theorems, we would add a new set of topics that have been explored using global techniques since Gravitation was published: wormholes and topological censorship; and closed timelike curves, chronology horizons, and chronology protection. See, for example, Everitt and Roman (2012) for a not very technical discussion with references to the most important literature or Friedman and Higuchi (2008) for greater technical detail.
   D. To Part VII we would also add the equivalent of one more chapter on The Physical Structure of Generic Singularities and the Interiors of Black Holes. This chapter would include the following.
   a. The material in Chapter 30 (3.C above) on the Kasner and Mixmaster solutions of Einstein's equation, plus a more detailed discussion of the Belinsky, Khalatnikov, and Lifshitz (BKL) analysis, which suggests there is a generic, spatially inhomogeneous variant of Mixmaster (p. 806 of Gravitation); also, a description of numerical relativity simulations (Garfinkle 2004; Lim et al. 2009) that prove this to be true and reveal some surprising twists missed by BKL.
   b. Analyses that show that the inner horizons, $r=r_{\text{,}}$$r=r_{\text{, }}$r=r_(", ")r=r_{\text {, }}$r=r_{\text{, }}$, of a Kerr or ReissnerNordstrøm black hole (Fig. 34.4) are highly unstable and that material or radiation falling into the hole triggers these instabilities, converting the inner horizons into generic null singularities (Poisson and Israel 1990; Marolf and Ori 2012).
5. In Part VIII, Gravitational Waves, the chapters on the theory of the waves and their generation are largely up to date, but the chapter on their detection is extremely out of date. More specifically:
   A. Chapter 35, Propagation of Gravitational Waves, is essentially up to date.
   B. The topics covered in Chapter 36, Generation of Gravitational Waves, are essentially up to date, but they need to be augmented by the following.
   a. An overview of gravitational wave sources that are likely to be observed in the next decade or two; see, for example, Buonanno and Sathyaprakash (2015); Creighton and Anderson (2011); or, for far greater detail, Maggiore (2017).
   b. A sketch of the post-Newtonian expansion of the waves from compact binary stars (higher-order corrections to this chapter's quadrupolar analysis); see, for example, Straumann (2013); and for greater detail, Poisson and Will (2014) or Blanchet (2014). Exercise 39.15 of Gravitation could be a starting point for this.
   c. A description of numerical relativity simulations of the inspiral and merger of black-hole binaries, and black-hole/neutron-star binaries, and their gravitational waves (e.g., Choptuik, Lehner, and Pretorius 2015; Maggiore 2017); also, the nonlinear dynamics of curved spacetime triggered by black-hole mergers (e.g., Owen et al. 2011; Scheel and Thorne 2013).
   d. A sketch of the analysis that shows that early-universe inflation parametrically amplifies gravitational vacuum fluctuations coming off the big bang, to produce a spectrum of primordial gravitational waves; see, for example, Mukhanov (2005) or Maggiore (2017).
   C. Chapter 37, Detection of Gravitational Waves, is highly out of date. Although nothing is wrong with this chapter, and it can be of conceptual value (particularly Sections 37.1-37.3), it focuses on vibrating mechanical detectors, which have largely been abandoned. So in a modern course, we would replace Sections 37.4-37.10 by the following.
   a. An overview of the four types of detectors that are expected to open up four different gravitational-wave frequency bands in the next two decades: ground-based interferometers, such as LIGO, which have already opened the high-frequency band ( $10-10,000\mathrm{Hz}$$10-10,000\mathrm{Hz}$10-10,000Hz10-10,000 \mathrm{~Hz}$10-10,000\mathrm{Hz}$ ); space-based detectors, such as LISA, in which drag-free spacecraft track each other with laser beams, that are expected to open up the low-frequency band (periods of minutes to hours) in the next 15 or 20 years; pulsar timing arrays (PTAs), which are expected to open the very low frequency band (periods of a year to a few tens of years) in the coming decade; and socalled B-mode polarization patterns in the CMB, which are induced by primordial gravitational waves with periods of millions to billions of years (the extremely low frequency band) and which may be definitively measured in the next decade or so. See, for example, Berger et al. (2015) and Maggiore (2017).
   b. Detailed analyses of (idealized) ground-based interferometers, spacebased detectors, and PTAs (e.g., Creighton and Anderson 2011; Saulson 2017; MCP); and analyses of the influence of gravitational waves on CMB polarization (e.g., Maggiore 2017).
   c. A summary of observations of gravitational waves, which in 2017 are solely those by LIGO, such as Abbott et al. (2016a).
   D. This could be a good spot, in an advanced course, to present an overview of what is known about the nonlinear dynamics of vacuum, curved spacetime (e.g., Scheel and Thorne 2014)-most of which has already been mentioned above.
   a. The chaotic spacetime dynamics near a generic Mixmaster (BKL) singularity (4.D.a).
   b. The more gentle dynamics near a generic null singularity (4.D.b).
   c. The phase transitions, critical behavior, and scaling that show up in (nongeneric) "critical" gravitational collapse; for example, Choptuik, Lehner, and Pretorius (2015).
   d. The interacting "tidal tendices" and "frame-drag vortices" that generate the gravitational waves in black-hole collisions; for example, MCP, or for greater detail Scheel and Thorne (2014), or for still greater detail Owen et al. (2011).
   e. Nonlinear, two-dimensional turbulence (energy cascades from small scales to large scales) triggered by mode-mode coupling in perturbations of a fast spinning black hole; see Yang, Zimmerman, and Lehner (2015).

ACKNOWLEDGMENTS

REFERENCES

PREFACE

ACKNOWLEDGMENTS

SPACETIME PHYSICS

CHAPTER

GEOMETRODYNAMICS IN BRIEF

§1.1. THE PARABLE OF THE APPLE

Once upon a time a student lay in a garden under an apple tree reflecting on the difference between Einstein's and Newton's views about gravity. He was startled by the fall of an apple nearby. As he looked at the apple, he noticed ants beginning to run along its surface (Figure 1.1). His curiosity aroused, he thought to investigate the principles of navigation followed by an ant. With his magnifying glass, he noted one track carefully, and, taking his knife, made a cut in the apple skin one mm above the track and another cut one mm below it. He peeled off the resulting little highway of skin and laid it out on the face of his book. The track ran as straight as a laser beam along this highway. No more economical path could the ant have found to cover the ten cm from start to end of that strip of skin. Any zigs and zags or even any smooth bend in the path on its way along the apple peel from starting point to end point would have increased its length.
"What a beautiful geodesic," the student commented.
His eye fell on two ants starting off from a common point $P$$P$PP$P$ in slightly different directions. Their routes happened to carry them through the region of the dimple at the top of the apple, one on each side of it. Each ant conscientiously pursued

§1.2. SPACETIME WITH AND WITHOUT COORDINATES

Now it came to me: . . . the independence of the gravitational acceleration from the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation. ... This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning.

Problem: how to measure in curved spacetime

Resolution: characterize events by what happens there
tack would record and reveal the shape of the surface. The precision could be made arbitrarily great by making the number of tacks arbitrarily large. It takes more daring to think of driving several score pitons into the towering iceberg. But with all the daring in the world, how is one to drive a nail into spacetime to mark a point? Happily, nature provides its own way to localize a point in spacetime, as Einstein was the first to emphasize. Characterize the point by what happens there! Give a point in spacetime the name "event." Where the event lies is defined as clearly and sharply as where two straws cross each other in a barn full of hay (Figure 1.2). To say that the event marks a collision of such and such a photon with such and such a particle is identification enough. The world lines of that photon and that particle are rooted in the past and stretch out into the future. They have a rich texture of connections with nearby world lines. These nearby world lines in turn are linked in a hundred ways with world lines more remote. How then does one tell the location of an event? Tell first what world lines participate in the event. Next follow each

is a convenient name for the vector "John" that reaches from
$$(x^0,x^1,x^2,x^3)=(77,23,64,11)$$$$\left(x^0,x^1,x^2,x^3\right)=(77,23,64,11)$$(x^(0),x^(1),x^(2),x^(3))=(77,23,64,11)\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(77,23,64,11)$$(x^0,x^1,x^2,x^3)=(77,23,64,11)$$
to
$$(x^0,x^1,x^2,x^3)=(78.2,22.1,64.0,13.1)$$$$\left(x^0,x^1,x^2,x^3\right)=(78.2,22.1,64.0,13.1)$$(x^(0),x^(1),x^(2),x^(3))=(78.2,22.1,64.0,13.1)\left(x^{0}, x^{1}, x^{2}, x^{3}\right)=(78.2,22.1,64.0,13.1)$$(x^0,x^1,x^2,x^3)=(78.2,22.1,64.0,13.1)$$

This kind of mathematical reasoning about dimensionality makes good sense at the everyday scale of distances, at atomic distances $\left({10}^{-8}\mathrm{cm}\right)$$\left({10}^{-8}\mathrm{cm}\right)$(10^(-8)(cm))\left(10^{-8} \mathrm{~cm}\right)$\left({10}^{-8}\mathrm{cm}\right)$, at nuclear dimensions $\left({10}^{-13}\mathrm{cm}\right)$$\left({10}^{-13}\mathrm{cm}\right)$(10^(-13)(cm))\left(10^{-13} \mathrm{~cm}\right)$\left({10}^{-13}\mathrm{cm}\right)$, and even at lengths smaller by several powers of ten, if one judges by the concord between prediction and observation in quantum electrodynamics at high

When $t=\mathrm{\infty }$$t=\mathrm{\infty }$t=oot=\infty$t=\mathrm{\infty }$, the new coordinate $\overline{t}$$\overline{t}$bar(t)\bar{t}$\overline{t}$ is $\overline{t}=\pi m$$\overline{t}=\pi m$bar(t)=pi m\bar{t}=\pi m$\overline{t}=\pi m$. The $r,\overline{t}$$r,\overline{t}$r, bar(t)r, \bar{t}$r,\overline{t}$ coordinates still provide no way to distinguish the points along the dotted line. They still give "cells in the egg crate" collapsed to zero content along the dotted line.

§1.3. WEIGHTLESSNESS

*Other perturbations had to be, and were, guarded against. (1) A bit of iron on the torsion balance as big as ${10}^{-3}\mathrm{cm}$${10}^{-3}\mathrm{cm}$10^(-3)cm10^{-3} \mathrm{~cm}${10}^{-3}\mathrm{cm}$ on a side would have contributed, in the Earth's magnetic field, a torque a hundred times greater than the measured torque. (2) The unequal pressure of radiation on the two sides of a mass would have produced an unacceptably large perturbation if the temperature difference between these two sides had exceeded ${10}^{-4}{}^{\circ }\mathrm{K}$${10}^{-4}{}^{\circ }\mathrm{K}$10^(-4)^(@)K10^{-4}{ }^{\circ} \mathrm{K}${10}^{-4}{}^{\circ }\mathrm{K}$. (3) Gas evolution from one side of a mass would have propelled it like a rocket. If the rate of evolution were as great as ${10}^{-8}\mathrm{g}/\mathrm{day}$${10}^{-8}\mathrm{g}/\mathrm{day}$10^(-8)g//day10^{-8} \mathrm{~g} / \mathrm{day}${10}^{-8}\mathrm{g}/\mathrm{day}$, the calculated force would have been $\sim {10}^{-7}\mathrm{g}\mathrm{cm}/{\sec }^2$$\sim {10}^{-7}\mathrm{g}\mathrm{cm}/{\sec }^2$∼10^(-7)gcm//sec^(2)\sim 10^{-7} \mathrm{~g} \mathrm{~cm} / \mathrm{sec}^{2}$\sim {10}^{-7}\mathrm{g}\mathrm{cm}/{\sec }^2$, enough to affect the measurements. (4) The rotation was measured with respect to the pier that supported the equipment. As a guarantee that this pier did not itself rotate, it was anchored to bed rock. (5) Electrostatic forces were eliminated; otherwise they would have perturbed the balance.

electrons [Witteborn and Fairbank (1967)] and individual mu mesons [Beall (1970)]. What is more, not one of these objects has to see out into space to know how to move.

Box 1.2 MATERIALS OF THE MOST DIVERSE COMPOSITION FALL WITH THE SAME ACCELERATION ('STANDARD WORLD LINE")